A circular disc of radius 'r' carries uniform charge distribution of surface charge density 'A'. If the disc lies on the z = 0 plane, with its axis along the z axis, find the electric field at a point P, along the z axis that passes through the center of the disk perpendicular to its plane, when z < 0.

To find the electric field at point P along the z-axis when z < 0, due to a circular disc with uniform charge distribution, we can use the concept of symmetry.

Here's how you can derive the electric field at point P:

1. Consider a small element on the surface of the disc, with an area dA. This small element can be considered as a point charge since it is very close to point P.

2. The electric field due to this small charge element at point P can be calculated using Coulomb's law as:

dE = (k * dq) / r²

where dE is the electric field due to the small charge element, dq is the charge on the small element, k is the Coulomb's constant (k = 9 × 10^9 Nm²/C²), and r is the distance between the small charge element and point P.

3. The total electric field at point P due to the entire disc can be obtained by integrating the electric field contribution from each small charge element on the disc's surface.

E = ∫ dE

4. Since the charge distribution on the disc is uniform, the charge on each small element can be expressed in terms of the surface charge density, A, and the area of the small element, dA, as:

dq = A * dA

where A is the surface charge density in C/m².

5. The distance between the small charge element and point P, r, can be calculated using the Pythagorean theorem as:

r = √(z² + R²)

where z is the coordinate of point P along the z-axis, and R is the radius of the disc.

6. Substituting the values of dq and r into the expression for dE, we get:

dE = (k * A * dA) / (z² + R²)

7. Now, we can substitute these values into the integral for the electric field:

E = ∫ dE = ∫ (k * A * dA) / (z² + R²)

8. To evaluate this integral, we need to determine the limits of integration for dA. Since we are considering the entire surface of the disc, the limits of integration for dA are from 0 to the total area of the disc, πR².

Therefore, the integral becomes:

E = ∫[0,πR²] (k * A * dA) / (z² + R²)

9. Evaluating this integral will give you the expression for the electric field at point P along the z-axis when z < 0.

E = (k * A * πR²) / (z² + R²)

Now you have the formula for the electric field at point P along the z-axis when z < 0, due to a circular disc with uniform charge distribution.